Advertisements
Advertisements
प्रश्न
If the first, second and last term of an A.P are a, b and 2a respectively, then its sum is
विकल्प
\[\frac{ab}{2 (b - a)}\]
\[\frac{ab}{b - a}\]
\[\frac{3 ab}{2 (b - a)}\]
none of these
Advertisements
उत्तर
\[\frac{3 ab}{2 (b - a)}\]
Let the A.P. be a, a+d, a+2d........a+nd.
Here, let d be the common difference and n be the total number of terms.
\[a_1 = a, \]
\[ a_2 = b\]
\[ \Rightarrow a + d = b\]
\[ \Rightarrow d = b - a . . . . . \left( 1 \right)\]
\[ a_n = 2a\]
\[ \Rightarrow a + \left( n - 1 \right)d = 2a\]
\[ \Rightarrow \left( n - 1 \right)d = a\]
\[ \Rightarrow d = \frac{a}{n - 1} . . . . . \left( 2 \right)\]
Given:
From equations \[\left( 1 \right) \text { and } \left( 2 \right),\] we have:
\[\Rightarrow \frac{a}{n - 1} = b - a\]
\[ \Rightarrow \frac{a}{b - a} + 1 = n\]
\[ \Rightarrow \frac{a + b - a}{b - a} = n\]
\[ \Rightarrow \frac{b}{b - a} = n\]
Now, sum of n terms of an A.P.:
\[S = \frac{n}{2}\left\{ a + a_n \right\}\]
\[ = \frac{n}{2}\left( 3a \right)\]
\[ = \frac{3ab}{2\left( b - a \right)}\]
APPEARS IN
संबंधित प्रश्न
If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference.
The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.
Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.
A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.
The Fibonacci sequence is defined by a1 = 1 = a2, an = an − 1 + an − 2 for n > 2
Find `(""^an +1)/(""^an")` for n = 1, 2, 3, 4, 5.
Find:
10th term of the A.P. 1, 4, 7, 10, ...
Find:
18th term of the A.P.
\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]
Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative term?
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.
If < an > is an A.P. such that \[\frac{a_4}{a_7} = \frac{2}{3}, \text { find }\frac{a_6}{a_8}\].
If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.
Find the sum of the following arithmetic progression :
50, 46, 42, ... to 10 terms
Find the sum of all integers between 84 and 719, which are multiples of 5.
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.
If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.
The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th terms.
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
a (b +c), b (c + a), c (a +b) are in A.P.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
bc, ca, ab are in A.P.
If a, b, c is in A.P., prove that:
a2 + c2 + 4ac = 2 (ab + bc + ca)
Show that x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P., if x, y and z are in A.P.
A manufacturer of radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find (i) the production in the first year (ii) the total product in 7 years and (iii) the product in the 10th year.
We know that the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
Write the common difference of an A.P. whose nth term is xn + y.
If m th term of an A.P. is n and nth term is m, then write its pth term.
If Sn denotes the sum of first n terms of an A.P. < an > such that
If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [cosec a1cosec a2 + cosec a1 cosec a3 + .... + cosec an − 1 cosec an] is
Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn − k Sn − 1 + Sn − 2 , then k =
If, S1 is the sum of an arithmetic progression of 'n' odd number of terms and S2 the sum of the terms of the series in odd places, then \[\frac{S_1}{S_2}\] =
Mark the correct alternative in the following question:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
The first three of four given numbers are in G.P. and their last three are in A.P. with common difference 6. If first and fourth numbers are equal, then the first number is
If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n
Find the rth term of an A.P. sum of whose first n terms is 2n + 3n2
If 100 times the 100th term of an A.P. with non zero common difference equals the 50 times its 50th term, then the 150th term of this A.P. is ______.
